[FOM] CH and forcing

[Mitchell Spector]

Prikry forcing provides an interesting example of internal forcing (inside V), 
since the models involved are in this case well-founded.  This can then be used 
to motivate a construction for the non-well-founded general case.

The usual approach to Prikry forcing would be to start with a ground model 
containing measurable cardinal kappa with normal ultrafilter U; a generic 
extension adds a new omega-sequence cofinal with kappa (while preserving 
cardinals, etc.).

An alternative approach ("internal forcing") takes place entirely within V:

Take iterated ultrapowers of V by U.  Because U is countably complete, the 
iterated ultrapowers are well-founded; each one is isomorphic to a transitive 
inner model.  Call the (Mostowski collapse of the) alpha-th iterate M_alpha. M_0 
here is V.  Let j_alpha_beta be the canonical elementary embedding which maps 
M_alpha to M_beta.

We'll use the class M_omega as the ground model.  (M_omega is an inner model of 
ZFC, and V can be elementarily embedded into it.)  The ordinal j_0_omega(kappa) 
is a measurable cardinal in M_omega, with normal ultrafilter j_0_omega(U).  But 
this ordinal is cofinal with omega in V.  The crux of the matter here is that, 
in fact, the sequence
<j_0_n(kappa) | n < omega>
is Prikry-generic over M-omega.  The generic extension of M_omega one gets from 
this is a transitive inner model (of V), and everything has taken place inside V.



One can use this as a template for a similar construction for any forcing 
notion, although the class models involved will usually not be well-founded. 
This general approach uses limit ultrapowers (of which iterated ultrapowers are 
a specific example).

I believe the following is correct, if memory serves:

Given a partial ordering P, the collection of dense open sets is a filter. 
Extend it to an ultrafilter U (using the axiom of choice).  Take the ultrapower 
V^P/U.  A function f with domain P is said to be eventually constant if the set
{ p | for all q extending p, f(p) = f(q) }
belongs to U.  Let M be the submodel of V^P/U consisting of the equivalence 
classes of the eventually constant functions.  This submodel M is a limit 
ultrapower of V.  (Chang & Keisler's Model Theory has a very clear exposition of 
limit ultrapowers.)

Let j be the canonical elementary embedding of V into M.  Then j(P) is a partial 
ordering in M.  Using M as the ground model, one can construct a generic 
extension of M by the forcing notion j(P).

Again, this takes place entirely in V.  (If we let G(p) = {q | p extends q}, 
then the generic extension is constructed as a submodel of the full ultrapower 
V^P/U, using [G] as the generic filter.  But the construction would need to be 
carried out in V itself, not in V^P/U, since the ground model M is not definable 
over V^P/U.)

Unlike the situation with Prikry forcing, the model M and its generic extension 
are, in general, not well-founded, and, as Andreas points out, typically not 
even omega-models.



I'm not familiar with Vopenka's result on this -- I'd like to thank Andreas 
Blass for posting his interesting comments about it.

The construction above comes from some old musings about ultrapowers and 
forcing.  Is this approach essentially the same as Vopenka's?  From Andreas' 
description, it sounds like it is, but of course using partial orderings instead 
of Boolean algebras.


I hope I've remembered all this correctly; it must be 20 years since I've 
thought about it.


Mitchell Spector





Andreas Blass wrote:
> I agree that it's not easy to find in the literature the fact that you can
> get two-valued but not-necessarily-well-founded models by first forming the
> usual Boolean-valued model and then dividing by an arbitrary (not necessarily
> generic) ultrafilter.  It was, however, published quite early by Vopenka; I
> believe it's in his very first papers about what he called the nabla model of
> set theory.  There's a very systematic treatment of this and many related
> topics in the book "Theory of Semisets" by Vopenka and Hajek.  Unfortunately,
> that book is not designed for browsing; you pretty much have to start from
> the beginning and read straight through in order to assimilate all the
> notational conventions. Perhaps I should add a word of caution about the
> models obtained in this way: Not only are they usually not well-founded,
> they're usually not even omega-models.   The "ground model" inside such a
> forcing extension (i.e., the extension of the predicate often written
> "V-check") is not isomorphic to the model you began with but rather to an
> elementary extension of it (specifically, a Boolean ultrapower).  In
> principle, this is no problem, but in practice it's easy to get confused,
> especially if one is accustomed to the usual picture, where one uses a
> generic ultrafilter and gets a copy of the original model as V-check in the
> extension.
>
> Andreas Blass
>
> On Jul 4, 2011, at 11:57 AM, Aatu Koskensilta wrote:
>
>> Quoting Andreas Blass<ablass at umich.edu>:
>>
>>> If one insists on two-valued models, the assertion is still correct,
>>> provided one does not require models to be well-founded.
>>
>> It's standard in expositions of forcing to restrict one's attention to
>> well-founded models. Since all the necessary inductions and recursions
>> involve only definable predicates this is clearly an overkill; all we need
>> is that the ground model satisfied these and those induction and recursion
>> principles (e.g. regularity given sufficient amount of replacement or
>> whatnot). And, as those in the know very well know, we can force over e.g.
>> uncountable models and obtain (sometimes necessarily) ill-founded models,
>> just by boneheadedly going through the formal motions.
>>
>> Is there in the literature any systematic, clear, motivated, account of
>> forcing covering all this? My personal experience is that everyone knows
>> these things, but I'd be hard pressed if someone asked for reference.
>>
>> -- Aatu Koskensilta (aatu.koskensilta at uta.fi)
>>
>> "Wovon man nicht sprechen kann, darüber muss man schweigen" - Ludwig
>> Wittgenstein, Tractatus Logico-Philosophicus
>>
>>
>
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